Can Anxious attachment & Avoidant attachment styles work?

I've been seeing a man for a couple of years who has Avoidant (fearful) attachment style tendencies (not wholly avoidant) while I am stable - but with Anxious tendencies.

That is to say, I'm ok with not hearing from him for a couple of days if he's busy - it doesn't send me into a frenzy. But when he is being most avoidant, I do find it difficult.

My question is, can these two attachment styles have a successful relationship or is it doomed by sheer virtue of the fact that they trigger dysfunctional behaviours in each other?